A112698 Partial sum of (Catalan numbers A000108 multiplied by powers of 4).

1, 5, 37, 357, 3941, 46949, 587621, 7616357, 101332837, 1375876965, 18987759461, 265554114405, 3755416368997, 53610591434597, 771525112379237, 11181285666076517, 163041321978836837, 2390321854565988197

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Fifth column (m=4) of triangle A112705.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-16*x])/(8*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
With[{nn=20},Accumulate[4^Range[0,nn] CatalanNumber[Range[0,nn]]]] (* Harvey P. Dale, Mar 11 2023 *)

PARI

x='x+O('x^50); Vec((1-sqrt(1-16*x))/(8*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

Formula

a(n) = Sum_{k=0,..,n} C(k)*4^k, n>=0, with C(n):=A000108(n).

G.f.: c(4*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

Recurrence: (n+1)*a(n) = (17*n-7)*a(n-1) - 8*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 16^(n+1)/(15*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

Extensions

Definition clarified by Harvey P. Dale, Mar 11 2023